1. 高エネルギーでのハドロン全断面積の 普遍的増加とＬＨＣでのｐｐ全断面積の予言
　　　 理研仁科センター　　　　　　　猪木慶治
　 ２０１１年３月９日　京都大学　基研研究会
　　　　　　素粒子物理学の進展２０１１
　　　　　　　Phys.Lett.B670(2009)395
　　　Phys.Rev.D79(2009)096003
　　　　 α
　　　　 In collaboration with Muneyuki Ishida

2. Outline of this talk Prediction of σ(pp) at 7TeV, 14TeV (LHC)
Test of Universality (Blog2s ) 　　　　 Is B universal between 2 body had. scatt.?
Answer :Yes
σ（＋）：parabola as function of logν
高エネルギーのデータを使って　B　を決めるのが普通、parabola左側の共鳴領域のデータをも使うことができ、Ｂ の決定の精度大。
Universality

3. 5. それをつかって、ＬＨＣにおいて、7TeV, 14TeV での予言。さらに、他のグループではできないπｐ、Ｋｐの予言。
6. 最後にＧＺＫエネルギー（Ｅ＝３３５ＴｅＶ）を含む超高エネルギーでの予言
　Auger 観測所との比較が楽しみ。

4. Test of Universality Increase of has been shown to be at most
by Froissart (1961) using Analyticity and Unitarity.
Soft Pomeron fit :　 Donnachie-Landshoff
σtot ~ s0.08 but violates unitarity
COMPETE collab.(PDG) further assumed
for all hadronic scattering to reduce the number of adjustable parameters based on the arguments of
CGC(Color Glass Condensate) of QCD. 4 4

5. σ ρ Particle Data Group (by COMPETE collab.) The upper side：σ
The lower side：ρ－ratio
B (Coeff. of
Assumed to be universal
Theory:Colour Glass Condensate of QCD sug. ρ
Not rigorously proved from QCD
Test of univ. of B:necessary
even empirically. 5 5

6. Ｉncreasing tot.c.s. Consider the crossing-even f.s.a. with We assume
　at high energies.
Increasing total cross sections in the previous graph, p.5
M : proton mass
ν, k : incident proton energy, momentum in the laboratory system 6

7. The ratio The ratio = the ratio of the real to imaginary part of 7
Since the amplitude is crossing-even, we obtain the above crossing relation. ReF/ImF in the graph of p.5. 7

8. How to predict σ and ρ for pp at LHC based on FESR duality?
(as example)
We searched for simultaneous best fit of σ and ρ up to some energy(e.g.,ISR) in terms of high-energy parameters constrained by FESR.
We then predicted and in the LHC regions. 8 8

9. FESR is used as constraint of
Both and data are fitted through two formulas simultaneously with FESR as a constraint.
FESR is used as constraint of
and the fitting is done by three parameters:
giving the least
Therefore, we can determine all the parameters
FESRによるconstraintにより、パラメーターが一つ減る。
These predict at higher energies including LHC energies. 9 9

10. The fit is done for data up to ISR
LHC
LHC
ISR(=2100GeV)
(a) All region
: High energy region
Predictions for and
The fit is done for data up to ISR
LHC
As shown by arrow.
(b) 10 10

11. Summary of Pred. for and at LHC
Predicted values of agree with pp exptl. data at cosmic-ray regions within errors.
It is very important to notice that energy range of predicted is several orders of magnitude larger than energy region of the input.
Now let us test the universality of B.
Root s < 62.7GeV(ISR) をインプットとしたときのまとめ。次からが Main Topic. 11 11

12. Main Topic:Universal Rise of σtot ?
B (coeff. of (log s/s0)2) :
Universal for all hadronic scatterings ?
Phenomenologically B is taken to be universal in the fit to πｐ,Ｋｐ, 　　,ｐｐ,∑ｐ,γｐ, γγ forward scatt.
　　　　 COMPETE collab. (adopted in Particle Data Group)
Theoretically Colour Glass Condensate of QCD suggests the B universality.
Ferreiro,Iancu,Itakura(KEK),McLerran(Head of TH group,RIKEN-BNL)’02
Not rigourously proved yet only from QCD.
 Test of Universality of B is Necessary even empirically. ｐｐ ー
　　　 Memo:　Therefore, it is worthwhile to prove or disprove this universality of B even empirically. 12 12

13. Particle Data Group (by COMPETE collab.) The upper side：σ
The lower side：ρ－ratio
B (Coeff. of
Assumed to be universal
Theory:Colour Glass Condensate of QCD sug.
Not rigorously proved from QCD
Test of univ. of B:necessary
even empirically. 13 13

14. through search for simultaneous best fit to experimental .
We attempt to obtain
for
through search for simultaneous best fit to experimental
Let us first attempt to obtain B values for pp, πp, Kp scatterings. 14 14

15. New Attempt for In near future, will be measured at LHC
energy. So, will be determined with good accuracy.
On the other hand, have been measured only up to k=610 GeV. So, one might doubt to obtain for (as well as ),
with reasonable accuracy.
We attack this problem in a new light.
pp よりroot s で二桁小さい。だから、pp のようによい精度で B の値を得られるかどうか疑問に思われるでしょう。 15

16. Practical Approach for search of B
Tot. cross sec.= Non-Reggeon comp. + Reggeon(P’) comp.
Non-Reg. comp. shows shape of parabola as a fn. of logν with a min.
Inf. of low-energy res. gives inf. on P’ term. Subtracting this P’ term from σtot(+), we can obtain the dash-dotted line(parabola).
Fig.1 pp, pp -
log ν(GeV) ●
Tevatron 　 s = 1.8 TeV √
SPS 　 s = 0.9 TeV
s = 60 GeV
ISR
σtot (mb)
100 50 40
Non-Regge  comp.(parabola)
Fig.1 pp, pp -
We have good data for large values of log ν compared with LHS, so (ISR, SPS, Tevatron)-data is most important for det. of c2(pp)(or Bpp) with good accuracy.
C2>0　故に上を向いた放物線；dash-dotted line: それにＰ’を加えたものが実線 16

17. Fig.2 πp - Fig.2 πp σtot measured only up to●
logν(GeV)
highest energy
s = 26.4 GeV
kL = 610 GeV √
σtot (mb)
Resonances ↓ 50 40 30
Non-Regge  comp(parabola)
Fig.2 πp
σtot measured only up to
s = 26.4 GeV (cf. with pp, pp). √ -
So, estimated Bπp
may have large uncertainty.
The πp has many res. at low energies, however.
So, inf. on LHS of parabola obtained by subtracting P’ term from σtot(+) is very helpful to obtain accurate value of B(πp).
(res. with k < 10GeV).
( Kp : similar to πp ) .
Pip: root s< 26 GeV two orders of magnitude smaller than root s< 1,8 TeV(=1800 GeV). 上の共鳴状態はFESR duality を通してＰ‘項の寄与を与え,parabola の LHS 項を与える。 17

18. Fitting high-energy datapp -
, pp Scattering
ISR
Ecm<63GeV
SPS
Ecm<0.9TeV
Tevatron
Ecm=1.8TeV
CDF D0
　σtot 　
Fitted energy region ｐｐ ー ー
For pp scatterings
We have data in TeV.
σtot = Bｐｐ (log s/s0)2 + Z (+ ρ trajectory)
in high-energies.
parabola of log s
Note: 横軸 root s
Bｐｐ = 0.273(19) mb
estimated accurately.
Depends the data with
the highest energy. (CDF D0)
　ｐｐ 18 18

19. πｐ , Ｋｐ Scatterings No Data No Data No Data in TeV
　πｐ
　π+ｐ
No Data
Ｋ-ｐ
Ｋ+ｐ
No Data
No Data in TeV
 Estimated Bπｐ , BＫｐ have large uncertainties. 19 19

20. Test of Universality of B
Highest energy of Experimetnal data:
　　 : Ecm = 0.9TeV SPS; 1.8TeV Tevatron
π-ｐ ： Ecm < 26.4GeV
Ｋｐ ： Ecm < 24.1GeV No data in TeV B : large errors.
Bｐｐ = 0.273(19) mb
Bπｐ = 0.411(73) mb  Bｐｐ =? Bπｐ =? BＫｐ　?
BＫｐ = 0.535(190) mb 　　 No definite conclusion 　　
It is impossible to test of Universality of B only by using data in high-energy regions.
We attack this problem using duality constraint from
　 ＦＥＳＲ(1): a kind of P’ sum rule ｐｐ ー 20 20

21. Kinematics ν : Laboratory energy of the incident particle
　　　　s =Ecm2 = 2Mν+M2+m2 ~ 2Mν
M : proton mass of the target. Crossing transf. ν  ー ν
m : mass of the incident particle
　　　m=mπ , mＫ , M for πｐ； Ｋｐ；ｐｐ; 　　　　　　　　　　　　　　　k = (ν2 – m2)1/2 : Laboratory momentum ~ ν
Forward scattering amplitudes　 fap(ν): a = ｐ,π+,Ｋ+
　　　　　Im fap (ν) = (k / 4 π) σtotap : optical theorem
Crossing relation for forward amplitudes:
f π-ｐ(-ν) = fπ+ｐ(ν)* , fＫ-ｐ(-ν) = fＫ+ｐ(ν)*
Notation of masses for incident particle for π, K, and p. このページは，さらりとやってスキップしてもよい。 21 21 21

22. Kinematics average of π-ｐ, π+ｐ； Ｋ-ｐ, Ｋ+ｐ； ｐｐ,
Crossing-even amplitudes : F(+)(ーν)=F(+)(ν)*
average of π-ｐ, π+ｐ； Ｋ-ｐ, Ｋ+ｐ； ｐｐ,
Im F(+)asymp(ν) = βＰ’ /m (ν/m)α’(0)
+(ν/m2)[ c0+c1log ν/m +c2(log ν/m)2]
βＰ’ term : P’trajecctory (f2(1275) ): αＰ’(0) ~ 0.5 : Regge Theory
c0,c1,c2 terms : corresponds to Z + B (log s/s0)2
　　　　　c2　 is directly related with B . (s～2M ν)
Crossing-odd amplitudes : F(-)(ーν)= ーF(-)(ν)*
Im F(-)asymp(ν) = βV /m (ν/m)αV(0) ρ-trajecctory:αV(0) ~0.5
βＰ’ , βV is Negligible to σtot( = 4π/k Im F(ν) ) in high energies. 22 22

23. ＦＥＳＲ(1) Ｄｕａｌｉｔｙ Remind that the P’ sum rule in the introd..
ＦＥＳＲ(1)　Ｄｕａｌｉｔｙ
Remind that the P’ sum rule in the introd..
Take two N’s(FESR1)
Taking their difference, we obtain
　　has open(meson) ch. below ,and div. above th.
If we choose to be fairly larger than we have no difficulty. ( : similar)
No such effects in
RHS is calculable from
The low-energy ext. of Im Fasymp.
LHS is estimated from
Low-energy exp.data.
N1 in resonance energy, N2 in asymptotically high energy.
Ppbar has open meson ch. below threshold. Even in that case this relation has physical meaning. 23

24. FESR(1) corresponds to n = -1 K.IGI.,PRL,9(1962)76
The following sum rule has to hold under the assumption that there is no sing.with
vac.q.n. except for Pomeron(P).
Evid.that this sum rule not hold  pred. of the traj. with
and the f meson was discovered on the
0.0015
-0.012
2.22
Igi-Matsuda: FESR, Dolen-Horn-Schmid; (n-th)moment sum rule n=1no
Igi-Matsuda:FESR, Dolen-Horn-Schmid;(n-th) moment sum rule n=1 を普通のFESRとすれば、n=-1 が上記、Ｐ’sum rule
VIP: The first paper which predicts high-energy from low-energy ( FESR1）
Moment sum rule において、n=-1 とおくと P’ sum rule に reduce. 24

25. in low-energy regions should coincide with
Average of
in low-energy regions should coincide with
the low-energy extension of the asymptotic
formula
This relation is used as a constraint between
high-energy parameters:
Very Important Point 25

26. Choice of N1 for πｐ Scattering
Many resonances Various values of N1
in π-ｐ & π+ｐ
The smaller N1 is taken,
the more accurate
c2 (and Bπｐ) obtained.
We take various N1
corresponding to peak and
dip positions of resonances.
(except for k=N1=0.475GeV)
For each N1,
FESR is derived. Fitting is performed. The results checked.
Δ(1232)
N(1520)
N(1650,75,80)
Note: data of pi(+)p is not shown in the figure.
Note: The data of pi(+)p are not shown in the above fig. -
Δ(1905,10,20)
Δ(1700) 26 26

27. N1 dependence of the result
N1(GeV) 10 7 5 4
3.02
2.035
1.476
c2(10-5)
142(21)
136(19)
132(18)
129(17)
124(16)
117(15)
116(14)
χtot2
149.05
149.35
149.65
149.93
150.44
151.25
151.38
N1(GeV)
0.9958
0.818
0.723
(0.475)
0.281
No SR
c2(10-5)
116(14)
121(13)
126(13)
(140(13))
121(12)
164(29)
χtot2
151.30
150.51
149.90
148.61
150.39
147.78
# of Data points : 162.
best-fitted c2 : very stable.
We choose N1=0.818GeV
as a representative.
Compared with the fit by
6 param fit with
No use of FESR(No SR)
Note: Error is large for No SR. 27 27

28. Result of the fit to σtotπｐ
No FESR
FESR used
FESR integral
Fitted region
Fitted region
π-ｐ
π+ｐ
much improved
c2=(164±29)・10-5 c2=(121±13)・10-5 Bπｐ＝0.411±0.073mb Bπｐ＝0.304±0.034mb 28 28

29. Result of the fit to σtotＫｐ
No FESR
FESR used
Fitted region
FESR integral
Fitted region
c2=(266±95)・ 　　 c2=(176±49)・10-4
BＫｐ＝0.535±0.190mb BＫｐ＝0.354±0.099mb
large uncertainty much improved 29 29

30. Result of the fit to σtotｐｐ,ｐｐー
No FESR
FESR used
FESR integral
Fitted region
large
Fitted region
large
The improvement in pp, pbarp is not remarkable,since high-energy data by ISR, SPS, which affects directly estimation of c2 are included in both fits.
c2=(491±34)・ 　　 c2=(504±26)・10-4
Bｐｐ＝0.273±0.019mb Bｐｐ＝0.280±0.015mb
Improvement is not remarkable in this case. 30 30

31. Test of the Universal Rise
σtot = B (log s/s0)2 + Z
B (mb) πｐ
0.304±0.034 Ｋｐ
0.354±0.099 ｐｐ
0.280±0.015
B(mb)
0.411±0.073
0.535±0.190
0.273±0.019
FESR used
Parabola で左辺に共鳴状態の寄与を入れたもの
No FESR
Bπｐ= Bｐｐ= BＫｐ within 1σ
Universality suggested.
Bπｐ ≠? Bｐｐ =? BＫｐ
No definite conclusion in this case. 31 31

32. Concluding Remarks In order to test the universal rise of σtot ,
we have analyzed π±ｐ;Ｋ±ｐ; 　 ,ｐｐ independently.
Rich information of low-energy scattering data constrain,　through FESR(1), the high-energy parameters B to fit experimental σtot and ρ ratios.
The values of B are estimated individually for three processes.
FESR means FESR(1). 32 32

33. We obtain Bπｐ= Bｐｐ= BＫｐ. Universality of B suggested.
Use of FESR is essential
to lead to this conclusion.
Universality of B suggests
gluon scatt. gives dominant cont. at very high energies( flav. ind. ).
It is also interesting to note that Z for
approx. satisfy ratio predicted by quark model. Ｋｐ πｐ ｐｐ
Within one standard deviation.
2:2:3 33 33

34. Our Conclusions at 7TeV, 14TeV will be tested by LHC TOTEM.
Our results
predicts at 7TeV
at 14TeV
　　　　　　　　　　　 at 335TeV
Our Conclusions at 7TeV, 14TeV
will be tested by LHC TOTEM.
Our Conclusions will be tested by LHC TOTEM.
VIP: GZK の場合は last graph を示した後に説明 34

35. Finally, let us compare our pred. at 14TeV with other pred. ref.
Ishida-Igi (this work)
Igi-Ishida (2005)
Block-Halzen (2005)
COMPETE (2002)
Landshoff (2007)
Pred. in various models have a wide range.
The LHC(TOTEM) will select correct one. 35 35

36. Very Important Point Ishida-Igi’approach gives predictions
not only for pp
but also for πp, Kp scatterings,
although experiments are not so easy
in the very near future.

37. Predictions for up to ultra-high energies including GZK
From Resonances
Cosmic rays
GZK
ν=6×1010GeV
Tevatron
SPS
最後にGZKを含むultra-high energyまでのpp全断面積の予言を示す。
LHC
Ecm=7, 14TeV
Non-Regge
comp.
(parabola)
Fitted energy region

38. Concluding Remarks の続き
この図からわかるように、入射陽子が宇宙背景輻射の光子と衝突してエネルギーを失うＧＺＫエネルギー（335TeV)でも、我々の予言の誤差は驚くほど小さくなっている。
Bの値は最高エネルギーのデータポイントの値に比較的強く依存する。 の値の予言のためにも、 LHCでの測定は大変重要。
また、LHCの測定で、CDF,D0のどちらが正しいかも決まる。

39. Appendices: FESR(1)
Define
We obtain

40. FESR(2)
Dolen-Horn-Schmid : (nth)-moment sum rule において、
n=1とおくと、FESR(2)
n= -1とおくと、FESR(1)=P’ sum rule(1962)